Density Operator

Let $H$ be a complex Hilbert space (typically finite-dimensional in basic quantum theory). A density operator (also called a density matrix) is an operator $\rho:H\to H$ such that:

1. Positivity: $\rho \ge 0$, meaning $\langle \psi,\rho\psi\rangle \ge 0$ for all $\psi\in H$.
2. Unit trace: $\operatorname{Tr}(\rho)=1$, where $\operatorname{Tr}$ is the operator trace (Trace Operator ).

In finite dimension, these conditions are equivalent to $\rho$ being a positive semidefinite matrix with trace $1$.

Basic structural facts (finite dimension)

• $\rho$ is automatically self-adjoint: $\rho=\rho^\ast$.
• All eigenvalues of $\rho$ are real and lie in $[0,1]$.
• The eigenvalues sum to $1$: if $\rho$ has eigenvalues $(p_i)$, then $\sum_i p_i=1$.

Thus $\rho$ admits a spectral decomposition

with $(\phi_i)$ orthonormal and $p_i\ge 0$, $\sum_i p_i=1$.

Pure vs mixed

• $\rho$ is a pure state iff it has rank $1$, equivalently iff $\rho^2=\rho$, equivalently iff $\operatorname{Tr}(\rho^2)=1$. (See Pure State Quantum .)
• Otherwise $\rho$ is mixed and can be written as a convex combination $\rho=\sum_k q_k |\psi_k\rangle\langle\psi_k|$ with $q_k\ge 0$, $\sum_k q_k=1$. (See Mixed State Quantum .)

Expectation values (Born rule in operator form)

If $A$ is an observable (a self-adjoint operator, see Self Adjoint Operator Observable ), then the expectation value in state $\rho$ is

This formula unifies pure and mixed states.

Dynamics and transformations (finite dimension)

• Unitary evolution: if $U$ is unitary, then $\rho$ evolves as $\rho \mapsto U\rho U^\ast$.
• Projective measurement: spectral projectors $P_i$ (as in Spectrum Self Adjoint Finite ) define outcome probabilities $\Pr(i)=\operatorname{Tr}(\rho P_i)$.

Information-theoretic quantities

Two common functionals of $\rho$ are:

• Von Neumann entropy: $S(\rho) = -\operatorname{Tr}(\rho\log\rho)$ (see Von Neumann Entropy ).
• Relative entropy: $D(\rho\|\sigma)=\operatorname{Tr}(\rho(\log\rho-\log\sigma))$ under suitable support conditions (see Quantum Relative Entropy ).